Magic Mirror on the Wall …

. . . who is the fairest one of all?

Now that we have (still tentatively) a fairness (C) measure that gives us a clear target to shoot for, 100%, let’s see how close we can get to that number. We’ll use the simulator, with the new fairness (C) as our magic mirror to see what design gets closest.

The ground rules are these: The contest is single-elimination. All designs have to have eight teams (which means there will be exactly seven matches). Skills are at random from at least half of a normal curve. Perfect seeding is available. Fairness (B) doesn’t count for anything, except that every player has to have some chance of winning the first prize. Any configuration of lines and rounds is acceptable. All brackets will be evaluated at luck = 1. Payout must be either winner-takes-all or 65%/35%.

Let’s start with a baseline for a perfectly ordinary bracket, with matches A1-A4, B1-B2, and C3. (I don’t have to draw this bracket for you, I trust). With blind draw and a 65/35 payout, we get f(C) = 72.58%.

We can do better. First, let’s go for a WTA payout. Good, we’re now up to 79.30%.

Let’s get some better players in here. Setting the elite threshold at zero (and thus drawing from only the top half of the normal distribution, we get up to 83.84%.

We can do better. Now let’s apply perfect seeding. Up to 86.51%. OK, now that we’ve given the top two teams their own halves of the bracket, let’s go back to 65/35 payouts so that they’ll both get paid more often. 86.97%.

That’s it – nothing else seems to help. So let’s play with the lines next. Learning from our shenanigans in search of the ugly bottom, let’s try a cascade:

This loads the dice in favor of the top seeds in a big way. Let’s load this bracket up with the things that worked before: Elite players, full seeding, and a 65/35 payout. Sure enough, that did the trick! Fairness (C) has now reached 95.34%. And nothing else seems to improve it. That’s a pretty darned fair bracket.

Time to send the simulator down to the marketing department so that they can repackage it as a magic mirror!